Calculus How To

Heaviside Function (Unit Step Function)

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What is the Heaviside Step Function?

The Heaviside step function (named after physicist Oliver Heaviside) is a simple discontinuous piecewise function defined over the interval (-∞, ∞). The function, usually denoted as H(t), equals:

  • 0 for all negative values of t,
  • 1 for all positive values of t.

Although the unit step function (a standard Heaviside, shown below) can only take on values of 0 or 1, it can be used to model other values. For example:

  • -9 μc(t) is a switch that turns on at time c with a value of -9,
  • 0.5 μc(t) is a switch that turns on at time c with a value of 0.5.

heaviside function graph

The graph, which is centered at zero, can be shifted along the x-axis to make a shifted Heaviside function. This is especially useful when modeling waveforms that are turned on or off at some interval other than t = 0.

Integral and Derivative

In terms of integrals, the Heaviside function is the integral of the Dirac function. The derivative of the Heaviside function is 0 for all x ≠ 0. At x = 0 the derivative is undefined. Although there isn’t a true derivative as such (i.e. one that is a function), the Dirac delta function can be used to approximate it.

Importance of the Heaviside Function

The Heaviside function is widely used in engineering applications and is often used to model physical systems in real time, especially those that change abruptly at certain times. For example, when current is turned on or off. In mathematics, the function is used as a basic building block with Laplace transforms, to shift functions.


Chapter 20: The Dirac Delta Function. Retrieved December 6, 2019 from:

Stephanie Glen. "Heaviside Function (Unit Step Function)" From Calculus for the rest of us!

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